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SYLLABUS 


COURSE  IN  THE  THEORY  OF  EQUATIONS, 


IIOSTON: 

I'UmjSIIKI)    liV    (UNN.    IlKAl'll.    &    CO. 

1  ••    / 


CopyRiniiTKD  Bv  s  I'U,  &  Go.,  1883. 


HALSTED 
HAEDY: 
HILL: 
PEIEOE ! 


WALDO 


Mathematical  Books. 


BYERLY :     Differential  Calculus  , 

Integral  Calculus  .     . 

Mensuration      .     , 

Elements  of  Quaternions 

Geometry  for  Beginners 

Elements  of  Logarithms 

Mathematical  Tables,  Chiefly  to  Pour  Pigures 

Three-  and  Pour-Place  Tables  of  Logarithms 

Multiplication  and  Division  Tables.    Polio  size, 

Small  size, 
WENTWOKTH i 

Elements  of  Algebra  ■     . 

Complete  Algebra  .     ,     ,     . 

Plane  G-eometry     .... 

Plane  and  Solid  Geometry    . 

Plane  Trigonometry.     Paper 

Plane  and  Spherical  Trigonometry 

Plane  Trigonometry  and  Tables.     Paper 
.  Plane  and  Spherical  Trigonometry  and  Tables 

Five-Place  Log.  and  Trig.  Tables.     Paper    . 

Cloth     . 
WENTWOETH  &  HILL : 

Practical  Arithmetic   ..■..,.. 
WHEELEE:  Plane  and  Spherical  Trigonometry      ,     .     . 


Intro. 
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m:\v  YORK.      (;hica«jo. 


OA 


SYLLABUS 


COURSE  IX  THE  THEORY  OF  EQUATIONS. 


oJOiOo- 


1.  Define  a  root  of  an  equation.  Explain  a  short  method  of 
substituting  any  given  number  for  x  in  a  numerical  equation, 
using  only  the  detached  coeflTicients. 

2.  If  a  is  a  root  of  the  equation,  the  first  member  is  divisil)le 
by  x  —  a.  Give  a  short  formal  pnjof.  Give  a  second  proof 
showing  the  form  of  the  quotient;  i.e.,  substitute  a  for  x  in  the 
first  nieml)er  of  the  equation,  and  then  subtract  the  result  from 
the  given  first  member,  and  the  equation  will  l)e  in  a  form  where 
every  term  is  obviously  divisible  b)'  x—a. 

3.  Prove  the  converse  of  the  theorem  in  §  2.  If  x  —  a  will 
divide  the  first  member  of  the  equation,  a  is  a  root, 

4.  Assuming  that  every  equation  has  at  least  one  root,  prove 
.that  every  equation  of  the  7ith  degree  has  n  roots  and  can  be 
thrown  into  the  form 

A  {x  —  a)  {x  -b)  {x-  c)  •••  =  0. 

5.  Show  that  the  coefficients  of  the  equation  are  simple  func- 
tions of  the  negatives  of  the  roots. 


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2 

G.  Show  that,  if,  on  substituting  two  different  vakies  for  x  in 
turn  in  the  first  member  of  an  equation,  the  results  are  of  oppo- 
site signs,  tliere  must  be  an  odd  number  of  roots  between  the 

values  substituted. 

« 

Solution  of  Numerical  Equations. 

Commensurahle  Hoots. 

7.  Prove  that  if  the  coefficients  of  an  equation  are  whole 
numbers,  and  the  coefficient  of  the  first  term  is  unity,  the  equa- 
tion cannot  have  a  fractional  root.  Any  commensural)le  root 
of  such  an  equation  must  then  be  an  exact  divisor  of  the  con- 
stant term  by  §  5.  ,   i 

8.  After  a  root  a  is  found,  the  degree  of  the  equation  may 
be  lowered  by  dividiug  by  x  —  a.  Explain  abbreviated  meth- 
ods of  division  :  first,  by  detached  coefficients  ;  second,  by  syn- 
thetic division. 

9.  Prove  the  theorem :  If  a  is  a  commensurable  root  it  will 
exactly  divide  the  constant  term,  the  quotient  thus  obtained 
plus  the  coefficient  of  the  preceding  term,  this  quotient  plus  the 
preceding  coefficient,  and  so  on,  and  the  last  quotient  will 
be  -1. 

Prove  the  converse  of  this  theorem. 

Give  the  working  method  of  finding  all  the  commensurable 
roots  of  the  equation  described  in  §  7. 

10.  If  the  coefficients  of  an  equation  are  whole  numbers,  and 
the  coefficient  of  the  first  term  is  not  unity,  show  that  the  equa- 
tion may  easily  be  transformed  into  one  where  the  coefficients 
are  whole  numbers  and  the  coefficient  of  the  fu'st  term  is  unity, 
and  may  then  be  treated  by  §  9. 


11.  Descartes'  Mule  of  Signs.  Explain  what  is  meant  b}-  a 
permanence  of  sign  ;  a  variation  of  sign. 

Prove  that  the  number  of  positiA'e  roots  of  an  equation,  com- 
plete or  incomplete,  cannot  exceed  the  number  of  variations  of 
sign.  Show  that,  b}'  reversing  the  signs  of  the  terms  of  odd 
degi-ee,  the  equation  may  be  transformed  into  one  whose  roots 
are  the  negatives  of  the  roots  of  the  given  equation,  and  that 
b}'  applying  Descartes'  Rule  to  the  transformed  equation,  infor- 
mation ma}'  be  obtained  concerning  the  negative  roots  of  the 
given  equation. 

Incommensurable  Boots.     Methods  of  Aj)j)roximation. 

12.  Explain  tlie  rough  laborious  method  of  api)roximating  to 
the  value  of  an  incommensurable  root  based  ii[)on  §  G,  and  show 
that  it  is  theoretically  capable  of  any  desired  degree  of  accu- 
racy, and  as  applicable  to  a  transcendental  as  to  au  algebraic 
equation. 

13.  Explain  briefly  Xeivton's  Method:  i.e..  find  a  jjortion  a 
of  the  root  by  {?  12;  let  .v  =  a+//,  and  substitute  this  value 
for  X  in  the  equation,  neglecting  higher  powers  of  //  than  the 
first,  thus  obtaining  an  equation  of  the  first  degree  to  deter- 
mine an  approximate  \alue  of  h.  The  result  is  reasonably 
accurate  if  h  is  small. 

14.  Horner's  Method.  Find  liy  §  12  a  portion  a  of  the  root. 
Transform  the  equation  into  one  whose  roots  are  less  by  a  than 
those  of  the  original  equation.  Treat  the  resulting  equation 
by  §  12. 

Sliow  that  the  coefficients  of  the  transformed  equation  will  be 
the  remainders  obtained  in  dividing  the  original  equation  repeat- 
edly by  X  —  a. 


Xewton's  Method,  §  13,  shows  that  an  approximate  A^alue  for 
the  rest  of  the  root  may  l)e  found  by  dividing  the  constant  term 
of  the  transformed  equation  b}-  tlie  preceding  coellicient. 

Describe  Horner's  Method  in  its  practical  abbreviated  work- 
ing form.     Show  how  to  deal  with  negative  roots. 

General  Methods  and   Theorems. 

15.  Explain  the  method  of  finding  equal  roots  by  obtaining 
the  greatest  common  divisor  of  the  first  member  of  the  equation 
and  its  derivative  with  respect  to  x. 

IG.    Describe  Sturm's  Functions.     Prove  Sturm's  Theorem. 

17.  In  dealing  with  a  given  numerical  equation  of  high  de- 
gree :  first,  test  for  commensurable  roots,  and  lower  the  degree 
of  the  equation  bj^  their  aid  if  any  are  found  ;  second,  test  for 
equal  roots  ;  third,  use  Horner's  Method  in  finding  apprcxi- 
matel}"  the  incommensurable  roots,  employing  Sturm's  Tiieorem 
as  an  auxiliary  if  it  proves  absolutely  necessary. 

TlWAGINARIES. 

18.  The  treatment  aud  use  of  imaginaries  is  purely  arbitrary 
and  conventional.  Define  the  square  root  of  —  1  as  a  symbol 
of  operation,  aud  state  the  conventions  adopted  to  govern  its  use. 

V  —  ft-  =  ft  V — 1 , 

{a  +  b)  V^  =  ft  V^  +  b  V^, 

ft V— 1 = V— 1 .  ft. 

Interpret  the  powers  of  V  — 1  by  the  aid  of  the  definition  and 
these  conventions. 

Show  that  these  conventions  enable  us  to  deal  with  imaginary- 
roots  of  a  quadratic,  and  that  their  treatment  and  properties 
are  closel}'  analogous  to  those  of  real  roots. 


19.  Show  why  a  +  ftV  — 1  is  taken  as  the  typical  form  of  an 
imaginary.  Explain  the  ordinary  geometrical  representation  of 
an  imaginary  b\-  the  position  of  a  point  in  a  plane.  N.B.  This 
interpretation  is  entirely  arbitrary,  but  has  proved  very  useful 
in  sugo'esting  important  relations  which  might  not  otherwise 
have  been  discovered. 

20.  Show  that  the  sum,  the  irroduct,  and  the  quotient  of  two 
imaginaries  are  imaginaries  of  the  topical  form. 

21.  Give  the  second  typical  form  of  an  imaginary  suggested 
b}'  the  graphical  construction  of  §  19. 

r(cos<^  4-  V  — 1  -sinc^). 

Define  the  modulus  and  the  argument  of  an  imaginary. 

State  the  convention  concerning  the  sign  of  the  modulus,  and 
show  that  the  argument  may  have  an  infinite  number  of  values 
differing  by  multiples  of  2  tt. 

22.  Show  that  the  modulus  of  the  product  of  two  imaginai'ies 
is  the  product  of  their  moduli,  and  that  the  argument  of  their 
product  is  the  sum  of  their  arguments.  Prove  the  theorems 
concerning  the  modulus  and  argument  of  the  quotient  of  two 
imaginaries  ;  of  a  power  of  an  imaginary  ;  of  a  root  of  an  imagi- 
nary. 

23.  Show  that  tlie  7ith  root  of  an}'  real  or  imaginary  has  n 
values,  having  the  same   modulus  and   arguments  differing  by 

multiples  of  — . 
n 

24.  Define  conjugate  imaginaries.  Prove  that  conjugate  im- 
aginaries have  a  real  sum  and  a  real  product. 

Show  that  if  an  equation  with  real  coefficients  has  an  imagi- 
nary root,  the  conjugate  of  that  root  is  also  a  root  of  the  equa- 
tion. 


6 

25.  Give  Cardan's  /Solution  of  a  Cubic  of  the  form 

x^  -\- qx -\- r  =  U. 

Consider  the  irreducible  case.    Give  a,  Trigonometric  Solution. 
Show  that  ail}'  cubic  cau  be  reduced  to  the  form 

or'  +  qx  +  r  =  0. 

Obtain  the  general  solution  of  any  cubic. 

26.  Give  Descartes'  a!nd  Euler's  Methods  of  solving  a  bi- 
quadratic equation. 

Symmetric  Functions  of  the  Roots  of  an  Equation. 

27.  Define  a  symmetric  function  of  several  quantities.     Show 
that  any  combination  of  symmetric  functions  is  symmetric. 

The  coefficients  of  an  equation  are  symmetric  functions  of  the 
roots  of  the  equation  by  §  5. 

28.  Explain   Newton's   Method   of   expressing  the  sums  of 
powers  of  the  roots  of  an  equation  in  terms  of  the  coefficients, 

fx  =  (x  —  a)  (x  —  b)  (x  —  c)  ••• 

Take  the  logarithm  of  each  member  and  differentiate 

fx  1  1  1 


fx       x  —  a      X  —  b      X  —  c 

Consider  the  case  where  the  required  power  is  less  than  the 
degree  of  the  equation  ;  where  the  required  power  is  greater  than 
the  degree  of  the  equation. 

29.  Give  the  short  practical  method  of  obtaining  the  sums  of 
powers  of  the  roots  of  a  numerical  equation.;  divide  xf'x  by  fx, 
and  the  coefficients  of  a-~\  x~^,  x~^,  etc.,  in  the  quotient  are 
Sj,  §2,  s,3,  etc.     Shorten  by  using  detached  coefficients. 


30.  Anj'  complicated  symmetric  function  can  be  made  to 
depend  upon  simpler  functions  so  that  only  rational  integral 
forms  need  be  specially  investigated. 

31.  Show  that  symmetric  functions  ma}-  be  expressed  in 
terms  of  the  sums  of  powers  of  the  quantities  involved. 

Consider  special  cases. 

32.  Explain  the  method  of  elimination  hij  the  aid  of  sym- 
metric functioufi. 

Deteuminants. 

33.  Show  that,  if   two  simultaneous    equations   of  the  first 

degree, 

«i.r  +  6,//  +  Ci  =  0, 

a.,x  -\-  h.,y  -f  c,  =  0, 

are  solved,  the  numerators  and  denominators  of  the  values  of  x 
and  >j  have  a  peculiar  symau'tric  finin. 

Explain  the  notation  adopted  for  writing  compactly  such 
expressions. 

Describe  a  Determituuit^  its  ro?/;s,  columns,  and  diagonal 
term. 

Give  a  rule  for  expanding  a  determinant.  Give  the  laiv  of 
signs. 

Illustrate  by  determinants  of  tiie  second  and  third  orders. 

34.  Show  that  a  determinant  may  be  broken  up  into  a  sum 
of  terms  each  im-olving  a  sub-determinant.     Illustrate. 

35.  Show  that  an  interchange  of  two  rows  or  of  two  columns 
will  change  the  sign  of  a  determinant. 


36.  Show  that  if  two  rows  or  two  cohnmis  are  kleutical,  the 
value  of  the  determinant  is  zero. 

37.  Show  that  if  each  constituent  of  anj^  row  or  column  is 
multiplied  by  a  given  quantity,  the  whole  determinant  is  nu;lti- 
plied  by  that  quantit}'. 

38.  Show  that  if  each  constituent  of  any  row  or  of  any  col- 
umn is  a  binomial,  the  determinant  can  be  broken  up  into  the 
sum  of  two  other  determinants  of  the  same  order. 

39.  Show  how  to  compute  the  value  of  a  numerical  determi- 
nant.    Consider  examples. 

40.  Explain  the  application  of  determinants  to  elimination  in 
the  case  of  n  equations  of  the  first  degree  between  n  unknown 
quantities. 

41.  Explain  the  application  of  determinants  to  elimination  in 
the  case  of  two  equations  of  any  degree  involving  two  unknown 
quantities. 

W.  E.  BYERLY, 

Professor  of  3Iathematics  in  Harvard  University. 


J.  S.  Gushing  &  Co.,  Printers,  Boston,  Mass. 


ELEMENTS  OF  THE  DIFFERENTIAL  CALCULUS. 

AVilli  Xniiurous  Examples  aud  Applicalious.  DciiitiiieU  for  Use  as  a  College  Text- 
Book.  IJy  AV.  E.  BvEuiiY,  Professor  of  Mathematics,  Harvard  University.  8vo. 
27;j  pages.     Mailiug  Trice,  §LMO  ;  Introduction,  $2.00. 

Tuis  book  embodies  the  resiilts  of  the  author's  exi)eiience  in 
teachiiio-  the  Calouhis  at  Coniell  and  Harvard  Universities,  and  is 
intended  for  a  text-book,  and  not  for  an  exhaustive  treati.se.  Its 
peculiarities  are  tlie  rigorous  use  of  the  Doctrine  of  l^iniits,  as  a 
foundation  of  tlie  subject,  and  as  pieliniinarv  to  the  adoption  of  the 
more  direct  and  ]>ractically  convenient  iiitinitesiniai  notation  and 
nomenclature;  the  early  introduction  of  a  few  simple  formulas  and 
methods  for  integrating;  a  rather  elaborate  treatment  of  the  use  of 
infinitesimals  in  pure  geometry ;  and  the  attempt  to  excite  and  keep 
up  the  interest  of  the  student  by  l)ringing  in  throughout  the  whole 
book,  and  not  merely  at  the  end,  numerous  applications  to  prac- 
tical prablems  in  geometry  ami  mechanics. 

Jamcs'MiUf'I't'iri't', /'ro/.  of  Afiith..  tilic  spirit,  and  calculated  to  develop  the 

flurvard  L'liircr^titj/  (From  the  Ilurrurd  same  ^llirit  in  the  learner.  .  .  .    The  book 

TtegUter):  In  inalliematics,   as   in   other  contains,  perhaj)s,  all  of  the  integral  cJiI- 

branches  of  study,  the  need  is  now  very  cidiis.  as  well  as  of  the  dlHerential,  that  is 

much  felt  of  leachhnj;   which   Is  treneral  necessary  to  the  ordinary  student.     And 

without  beini?  siii)erticial ;  limited  to  lead-  with    so    ntuch   of    this    great  acicutific 

ing  tojdcs,  aud  yet  within  Its  limits:  tlior-  method,  every  thorough  student  of  phy- 

ougli,   accurate,   and '  jiractical  ;   adapted  sics,  and  every  general  scholar  who  feels 

to  the  comiiiunieatiiin  of  some  degree  of  any  interest  in  the  relations  of  abstract 

power,  as  w«'Il   iis   knowledg;',   but  free  thought,   and   f«  capable   of  grasping    a 

from  details  which  are  im|K>rtant  only  to  mathematical  i<lea,  ought  to  be  familiar, 

the   specialist.      Prof,    llyorly's  Calculus  One   who  aspires   to   technical    leaniin^ 

appears  to  be  designed  to  meet  this  want,  must  siiiii)leinent  his. mastery  of  the  ele- 

...    Such  a  plan   leaves   much   room   tor  menls  by  the  study  of  the  comprehensive 

the  <^xercise  of  individual  judgment;  and  theon-tiial   treatises.  .  .  .     Hut  lie  who  is 

differences  of  opinion   will  imdoubledly  thorotighly  acquainted  with  the  book  be- 

exist  in  regard   to  one  and  another  iM>int  fore   us   has   made  a  long  stride  into  a 

of  this  bi)ok.     But  all  teachers  will  agree  sound   anri    prai-lical    knowledge   of   the 

that  in  selection,  arrangement,  anil  treat-  subject  of  the  calculus.     He  has  begun  to 

menl,  it  is,  on   the  whole,  in  a  very  liiu'h  biva  real  analyst. 
degree,  wise,  able,  marked  by  a  true  si-icii- 


ELEMENTS  OF  THE  INTEGRAL  CALCULUS. 

With  NuMieriius  Kxampli-.-^  aiul  .\ppllcjiIions  ;  containing  a  Cliajiler  on  the  Calculus 
of  Imauinaries.  am!  :i  Practical  Key  to  the  Solution  of  Differential  P^quations. 
l)esi';nc.l  for  Use  .is  a  College Text-iSouk.  15y  W.  K.  I'.VKRI.Y.  Prof,  of  Mathefnatics 
in  llarvaril  Uuiversity.    Syo.    "ilU  ])agi-s.    Mailing  I'ricc,  sJ'.M.'i;  Introduction,  $2.00. 

This  volume  is  asetpiel  to  the  author's  t't-atise  on  the  Differential 
Calculus,  and,  like  that,  is  written  as  a  text-book.  The  last  chap- 
ter, however.  —  a  Key  to  the  Solution  of  Differential  Etjuations, — 
may  prove  of  .service  to  working  mathematicians. 

H.    \.  Newton,    Prxfi'xaiT   of  Mathematics.     Yith-  <\.n,iiv  ■    \\\.    k1,-,ii    ,,,..    it    i,, 
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